Preprint typeset in JHEP style - HYPER VERSION Topological Superconductor: A Review Theoretically & Experimentally Ze-Yang Li 1,2† , 1300011317 1 State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University; Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China 2 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, P. R. China † [email protected]Abstract: There is a saying that theoretical constructions never go beyond experimen- tal works in superconductor field. However, with recent year’s tremendous development of topological material, scientists realized that topological states of matter can actually be widespread, and hence the idea of topological superconductor became much more in- teresting and hence numerous theoretical works within the framework of well-established description of topological material seems to have a chance to gain the opportunity to get a step forward. Recent experiments, which in great agreement with theoretical predic- tions, confirm that. In this review, we first briefly review topological insulators’ theory and get some intuition what topology state means, and then construct basic theoretical description of topological superconductors with consideration of Majorana fermions, then finally briefly analyze and discuss the current status and further perspective of topological superconductors, both experimentally and theoretically. Keywords: Topological Superconductors, Majorana Fermions, Topological Material.
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Preprint typeset in JHEP style - HYPER VERSION
Topological Superconductor: A Review Theoretically
& Experimentally
Ze-Yang Li 1,2†, 1300011317
1State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University;
Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China2International Center for Quantum Materials, School of Physics, Peking University,
1. Introduction and A Quick Review of Topological Insulator Theory 2
1.1 Berry Phase, Quantum Spin Hall Effect as the Inspiration of Topological
Materials 2
1.2 Theoretical Description for Topological Insulator 6
2. Basic Theoretical Model 7
2.1 Introducing the Bogoliubov-de-Gennes (BdG) Formalism for s-Wave Super-
conductors 8
2.2 Majorana Description for a Condensed Matter System 8
2.3 Why p-wave Superconductors are Topologically Non-trivial 10
3. Random-Matrix-Theory (RMT) Approach 12
4. Experimental Confirmation 12
4.1 [5]’s Science: Signatures of majorana fermions in hybrid superconductor-
semiconductor nanowire devices 12
4.2 [6]: Zero-bias peaks and splitting in an Al-InAs nanowire topological super-
conductor as a signature of Majorana fermions 14
4.3 [7]: Observation of superconductivity induced by a point contact on 3D
Dirac semimetal Cd3As2 crystals 16
4.4 Fractional Josephson Effect 17
5. Qubit based on Majorana Fermions: a Start 17
6. Perspective 18
7. Acknowledgements 18
Appendix A 2-state Berry Phase Calculation I
– 1 –
Ze-YangLi’sAssignment(Draft)
1. Introduction and A Quick Review of Topological Insulator Theory
With the explosion of interest in unconventional superconductivity in the past two decades,
there have been two primary research foci: (1) the microscopic mechanism that produces
the unconventional superconducting pairing potential and (2) new quasiparticle phenomena
(Bogoliubov-like excitation) [1].
As well demonstrated in the course presentation, the theoretical description of topo-
logical material goes first at quantum spin hall effect. Actually, quantum spin hall effect
can be regarded as a two-dimensional version of topological insulator for only its edge
state has non-zero conductance contribution while its bulk state doesn’t. Like our starting
expectation of topological material, quantum spin hall effect gives a dissipationless trans-
port in both side both direction. It’s strong against impurities due to the additional berry
phase introduced by the impurity just interference to cancel backward scattering. In this
subsection, we first review what’s berry phase, followed by Bernivig’s book [1], and then
construct theoretical model for QSH/TI, then show the computational simulation of such
systems.
1.1 Berry Phase, Quantum Spin Hall Effect as the Inspiration of Topological
Materials
Berry phase is a geometry phase accumulated when the Hamiltonian is slowly changed in
the space of parameter. We consider a adiabatic dynamical evolution under a parametric
Hamiltonian H(R) on its n-th eigenstate, for a closed loop in parameter space:
|n(R(t))〉 = eiθ|n(R(0))〉,
where the additional phase is partially due to dynamical term,
θ =1
~
∫ t
0En(R(t′))dt′ − i
∫ t
0
⟨n(R(t′))
∣∣∣∣ ddt′∣∣∣∣n(R(t′))
⟩
An additional topological term comes from the second term if the system is topological
nontrivial. A straight forward way of such nontrivial Hamiltonian is H(R) = ε(R) + R ·σfor a 2-state system. (which would be regarded as an example at Appendix A).
In general, as we transform the time-differentiate to parameter-differentiate, i.e., d/dt⇒∇RR, we have the additional geometry phase can be calculated through a path in param-
while these operator also have “real-valuedness” and
obey anti-commutation relation. Fermion number is
n1 = (1− iγ2γ1)/2, n2 = (1− iγ4γ3)/2.
Hence it’s really straightforward to generalize to N fermion mode ψ1, · · · , ψN , basis
are |n1, n2, · · · , nN 〉, with the 2N Majorana operator:
γ2i−1 = ψi + ψ†i = σ3 ⊗ · · · ⊗ σ3(i−1)⊗σ3
⊗ σ1 ⊗ σ0 ⊗ · · · ⊗ σ0(N−i)⊗σ0
(2.3)
γ2i = i(ψi − ψ†i ) = σ3 ⊗ · · · ⊗ σ3(i−1)⊗σ3
⊗ σ2 ⊗ σ0 ⊗ · · · ⊗ σ0(N−i)⊗σ0
(2.4)
The property is also similar:
• “real-valuedness” γi = γ†i .
• Anti-commutation γi, γj = 2δ − ij.
– No vacuum for majorana while γγ = 1 s.t. There is no non-trivial bosonic
hermitian operator from a single Majorana, and
γγ|·〉 = |·〉 6= 0⇒ |·〉 6= 0
• Particle-hole transformation of fermion if switch two γ operator:
ψi =γ2i−1 + iγ2i
2→ γ2i + iγ2i−1
2= iψ†
• Non-locality:
– γj affects/depends on many sub-Hilbert spaces.
– Hamiltonian can only contain products of even number of Majorana fermion
operators.
– Hamiltonian preserves fermion parity
[P, H] = 0, P = (−1)∑i ni
– 9 –
Ze-YangLi’sAssignment(Draft)
– Non-trivial observables must contain two or more Majoranas (information is
stored non-locally).
• Non-Abelian statistics
– Abelian statistics: with certain number of fermions at fixed positions
∗ Hilbert space is 1-D.
∗ Eexchanges of fermion pairs just change the phase of wavefunction. Different
fermion pair exchanges commute.
– Non-Abelian statistics: with 2N Majoranas at fixed positions,
∗ Hilbert space is 2N -D
∗ Ddifferent Majorana pair exchange/braiding do not commute: represented
as non-commuting 2N × 2N matrices.
– Braiding of Majorana fermion: a unitary transformation on Hilbert spaceρ[σi,j ] :
γi → γj , γj → −γi.
ρ[σij ] =1− γiγj√
2
– Example: ρ[σ2,3]ρ[σ1,2] 6= ρ[σ1,2]ρ[σ2,3], see Figure 6.
Our goal is to realize well-separated localized Majorana zero
Figure 6: Non-Abelian
Braiding
modes in a system with bulk gap. We know the Majorana zero
modes that [γ,H] = 0. Actions of these Majoranas do not change
energy. If we have 2n Majorana zero modes we have 2n-fold de-
generate ground states. Majorana zero modes act non-trivially in
this subspace. Clearly, it requires bulk gap so that there is clear
separation between ground & excited states. Localized well well
separated so that the local perturbations will not lift the “topo-
logically protected” ground state degeneracy for it cannot involve more than one Majorana
mode.
For detailed modeling, we can see the next part.
2.3 Why p-wave Superconductors are Topologically Non-trivial
Here we consider the 1-D p-wave superconductor. It might sound weird about the idea
of 1-D superconductors; however it’s okay for the boundary - the end of a chain - has a
gapless dispersion.
We use the following parameter to describe: t the hopping energy, µ the chemical
potential, N the number of site (assumed to be even) and ∆ the p-wave interaction. Based
on this, we can write down the Hamiltonian like
H = H0 −∆
N−1∑i
(ψiψi+1 + ψ†i+1ψ†i ), H0 = −t
N−1∑i
(ψ†iψi+1 + ψ†i+1ψi)− µN∑i
(ψ†iψi −1
2)
(2.5)
– 10 –
Ze-YangLi’sAssignment(Draft)
(a)
(b)
(c)
Figure 7: 1-D Majorana illustration. (a) the general case, (b) trivial phase for local coupling, (c)
non-trivial for distance coupling
In terms of Majoranas, it becomes a tight-bind model of Majorana fermions. This is exact
the 1-D Kitaev Chain.
H =i
2
N−1∑i
[(∆− t)γ2i+1γ2i + (∆ + t)γ2i+2γ2i−1] + µN−1∑i
γ2iγ2i−1
(2.6)
There are two solvable special case. First , t = ∆ = 0, µ < 0, the Hamiltonian
is a simple sum over N mutally commuting terms and has a unique ground state, i.e.,
∀i, ψ†iψi = 0. The bulk excitations of energy is −µ when iγ2iγ2i−1 = −1. Second ,
t = −∆ > 0, µ = 0.1. In this case, H = −t∑N−1
i iγ2i+1γ2i. It’s a sum over N −1 mutually
commuting terms iγ2i+1γ2i. Ground state is all iγ2i+1γ2i = 1, and bulk excitations of
energy wt is one iγ2i+1γ2i = −1. Hamiltonian is not relates to γ1, γ2N , i.e., the Majorana
zero mode exists: there exists a Two-fold degeneracy: iγ1γ2N = ∓1 . Action of γ1(2N)
switches between the two degenerate states.
– 11 –
Ze-YangLi’sAssignment(Draft)
In this case, the fermion parity is either 1/0. For
εk
∆k
εk
∆k
(a)
(b)
|µ| < |2t|, nontrivial
|µ| > |2t|, trivial
Figure 8: Illustration of trivial or
nontrivial case for odd/even winding
around (0, 0.
a intuitive consideration, the ground state degeneracy
change from 1 to 2 is really nontrivial. Use the idea of
previously mentioned BdG form,
H =1
2
∑k
(ψ†k, ψ−k)(εkσz + ∆kσy)
(ψkψ†−k
)
with εk = −µ− 2t cos k,∆k = 2∆ sin k, the dispersion
is simply E(k) = ±√ε2k + ∆2
k. Under a mapping from
k → (εk,∆k), the image of Brillouin zone is a closed
loop; whether it contains the (0, 0) is the distinction of
trivial or not. I.e., winding around the origin odd(non-
trivial) or even(trivial) number of times.
This universal idea is so far the most widely used
way to achieve Majorana fermion. For experiment, see
section 4
3. Random-Matrix-Theory (RMT) Approach
The application of RMT to superconductivity is based on a connection between the quasi-
particle excitation spectrum and the eigenvalues of a real antisymmetric matrix; for a
review, see [4]. Due to the limit of time, this part is omitted (and might could be found at
my website laserroger.github.io/LaserPublic/academic_research.html
4. Experimental Confirmation
We briefly take a look at three fabulous experiments: [5], [6] and [7].
4.1 [5]’s Science: Signatures of majorana fermions in hybrid superconductor-
semiconductor nanowire devices
This part we talk about the experimental work by Mourik et at. It just the same as what we
mentioned at previously Majorana fermion consideration that realize a 1-D spinless fermion
with p-wave pairing. The approach is semiconductor wire with spin-orbital coupling plus
Zeeman field and proximity to s wave superconductor. The energy band after soc term is
no-longer degenerate, for the orbital part is strongly related to the orbital part and hence
the different spin index represents different band and has only a degeneracy point at zero
momentum; with additional Zeeman field, the degeneracy is lifted further and is gaped at
all momentum space, as in Figure 9.
This method is proposed at Fu Liang & Xu Cenke’s work [9]. Edge states of 2D topo-
logical insulator (i.e., Quantum Spin Hall system) is to some extent in proximity of s-wave
superconductor. They described it as a novel superconductor-ferromagnet-superconductor
(SC-FM-SC) Josephson array deposited on top of a two-dimensional quantum spin Hall
insulator. The observable phenomenon rather than intuition is that The Majorana bound
– 12 –
Ze-YangLi’sAssignment(Draft)
Figure 9: The energy band shaping by SOC and Zeeman field.
Figure 10: Braiding/exchange in “1D” without “collision” by sidetracks.
state at the interface between SC and FM leads to charge-e tunneling between neighboring
superconductor islands, in addition to the usual charge-2e Cooper pair tunneling. That’s
exactly the Majorana fermion phenomena that one can consider Majorana fermion as ‘half
a fermion’. More specifically, it’s a braiding/exchange in “1D” without “collision” by side-
tracks. By gating the local µ, trivial large µ to nontrivial small µ transport is derived, see
Figure 10.
The transport measurement can show
Figure 12: Color-scale plot of dI/dV versus V and
B. The ZBP is highlighted by a dashed oval; green
dashed lines indicate the gap edges. At ∼ 0.6T ,
a non-Majorana state is crossing zero bias with a
slope equal to ∼ 3meV/T (indicated by sloped yel-
low dotted lines).
the phase transition in the bulk super-
conductor. When there is a transition
from the trivial to the topological non-
trivial superconducting states by turning
on a magnetic field, the bulk gap would
collapse and then reopen as the field strength
increase would provide strong but indi-
rect evidence for the onset of topologi-
cal superconductivity and, by extension,
the appearance of Majorana and states in
the wire, see Figure 12. In (a), the triv-
ial case, only available low-energy degree
of freedom reside in the metallic region
lie along x < 0. Using the Green func-
tion and S matrix method for scattering
in transport experiment, the differential
conductance G(V ) = dI/dV is derived
by
– 13 –
Ze-YangLi’sAssignment(Draft)
G(V ) =2e2
h|SPH(eV )|2 (4.1)
where |SPH(eV )|2 is the probability that an incident electron at energy E Andreev reflects
as a hole at the junction, passing charge 2e into the superconductor, a bias voltage V
applied across the junction generates a current. However, particle-hole symmetry of the
modeling Hamiltonian (BdG) shows that
S(E) = σxS∗(−E)σx
requires S(0) to be completely diagonal or completely ofF-Diagonal. The purely diagonal
case corresponds to the onset of perfect normal reflection?with unit probability an electron
reflects as an electron and similarly for holes – and hence a vanishing zero-bias conductance.
In contrast, the ofF-Diagonal possibility yields perfect Andreev reflection; here electrons
scatter perfectly into holes and vice versa, yielding a quantized zero-bias conductance.
4.2 [6]: Zero-bias peaks and splitting in an Al-InAs nanowire topological su-
perconductor as a signature of Majorana fermions
This Nature Physics paper is much more clear in delivering its information. It contains not
only a precise introduction of TSC, Majorana fermion, but also a good numerical method.
Considering this paper also measure Zero Bias Conductance, I here mainly express how
this universal paper is derived based on numerical simulation.
While the system is likely to be in the diffusive regime, we work with a model of an
infinite ballistic wire divided into M segments, each segment is described by a 4× 4 BdG
Hamiltonian. The Hamiltonians of the segments depend on three parameters µ,Ez,∆. For
example, a segment with ∆ = 0 represents a normal wire (such as the leads of the device),
and a segment with Ez > ∆ and µ = 0 represents a topological superconductor. Total
Hamiltonian is H(x, p) =∑L
l=1 Πl(x)Hl(p). Πl(x) is a boxcar function which equals when is
inside segment l and zero otherwise (segments 1 and M are semi-infinite). Given an energy
E , we find the momenta pElm of the modes in each segment, as well as the corresponding
eigenvectors:
H(pElm)vElm = EvElm (4.2)
After discarding divergent modes in the left-most and right-most segments, we write
a wavefunction of a scattering state which is a general linear combination of the modes in
each segment (the index j goes over all the scattering states):
wEj (x) =
∑l,m
Πl(x)aEjlmvElme
ipElmx
We then determine the coefficients aEjlm by requiring that the wavefunction and its
derivative are continuous at the interfaces between the segments (we can also add barriers
– 14 –
Ze-YangLi’sAssignment(Draft)
Figure 13: Comparison between experiments(Left) and simulations(Right). Left: Color plot of
the ZBP with equal height contours lines, from 0.106e2/h to 0.197e2/h. The arrows indicate the
transition from a single ZBP to split peaks. Right: Simulated behaviour using analytical expressions
for the wire spectrum. Contours lines of constant-size Majorana wavefunction, ξ = hvF /Eg ∼ 1.5L,
3L and 10L are blue, red and black, respectively. The simulation of ξ < 3L (red line contour) is
similar to the contours of the data Left
in the form of delta-function potentials, which will impose jumps in the wavefunction
derivative). The modes with real momenta and positive (negative) group velocity in the
left-most (right-most) segment are incoming modes, and it turns out that choosing the
coefficients of the incoming modes determines the coefficients of the rest of the modes. We
can therefore choose a basis for the scattering states such that each basis state will have
only one incoming mode, and separate them into right-incoming and left-incoming states.
We normalize the scattering states such that:
〈wEj |wE′
k 〉 = δ(E − E′)δjk
The charge density and charge current carried by a scattering state:
ρ[w] = ew†τzw,J[w] = eRe[w†( pm
+ αsocσzτz
)w]
where αsoc is the soc constant. The charge continuity equation gives
∂ρ[w]
∂t+ ∇ · J[w] = 4e∆Im[w†σxτxw]
From this equation we see that there is a source (or drain) term due to the supercon-
ductor (where ∆ 6= 0). This is an important property of the BTK model, which implicitly
accounts for the current that goes through the superconductor and into the ground. To
obtain the current we need to integrate over the energy, using the Fermi-Dirac distribution
as the occupation:
I =
∫dE
fF-D(E − eV )∑j∈inc
J [wEk ]
– 15 –
Ze-YangLi’sAssignment(Draft)
Herewehaveattributedtheentirevoltagedroptotheelectronsincomingfromtheleft. It is also
possible to divide the voltage drop differently between electrons incoming from left and
right, to better capture the real capacitance relations of the contacts. The differential
conductance is:
G =dI
dV= −e
∫dEf ′F-D(E − eV )
∑j∈left-inc
J [wEk ]
To account for the appearance of the
Figure 14: (a) Basic experimental setup re-
quired to observe the fractional Josephson effect
stemming from Majorana modes fused across a
superconductor-insulator-superconductor junction.
Purple regions indicate s-wave superconductors that
drive the green regions into a 1D topological state;
dashed regions are trivially gapped. The Majorana
γ1,2 mediate a component of the Josephson current
that is 4π periodic in φR − φL. When the barrier
in the junction is replaced by a superconductor with
phaseφM as in (b), two Majoranas mediate a second
type of unconventional current that is 4π periodic
in both side and can be isolated with Shapiro step
measurements, while in (c), yields only conventional
Josephson physics with 2π periodicity.
two superconducting gaps (∆ind,∆Al), we
have added up the conductance of two
different structures. The main channel
is simulated by a wire with a topological
segment that has the proximity-induced
pairing potential of ∆ind. A second chan-
nel is simulated by a wire which ends in
a superconducting segment with the Alu-
minum pairing potential of ∆Al.
This two-channel technic is a prac-
tical way to simulate the main features
of the experiment within the limitations
of the numerical model. In the exper-
imental devices (when one of the elec-
trodes is pinched off) all the electrons
eventually go into the Aluminum, and
when their energy is below the Aluminum
superconducting gap, do so almost ex-
clusively by Andreev reflections. To ac-
count for this we only take the contribu-
tion of Andreev reflection into account
when calculating the conductance in the
main channel.
4.3 [7]: Observation of superconductivity induced by a point contact on 3D
Dirac semimetal Cd3As2 crystals
This paper mainly consider the new proposed Cd3As2 Dirac semimetal crystals. Using
point-contact spectroscopy measurements, exotic superconductivity around the point is
obtained and the ZBCP is derived which could be a signal for topological p-wave supercon-
ductor. The experiment is also similar, but the different is the material and the method.
The superconductivity is derived via a point contact equipment, which is also called the
‘needle-anvil’ configuration. This group proposed an theoretical explanation of the exotic
feature based TSC. Personally, I think this is a first step to 3-D TSC material, though still
within the transport measurement method.
– 16 –
Ze-YangLi’sAssignment(Draft)
Figure 15: a, Point Contact resistance versus temperature curve of Sample 2, showing an onset
Tc of 7.1K. Inset: Temperature dependence of the bulk resistivity measured by the standard
four-probe method, showing non-superconducting behaviour. b, Normalized PC spectrum at a PC
resistance of 65 for temperatures from 0.28 to 4.0K. c, ZBCP at different temperatures
4.4 Fractional Josephson Effect
This design contains a trivial gap separating two topological superconductors. Cooper
pairs and Majoranas aside the gap both contribute to the conducting current I = IC + IMwhich is different in periodic to the SC phase due to different charge carry. Considering
the difficulties to detect, the only possible way is SC-Insulator-SC junction, as illustrated
in Figure 14. Due to the passage limit, the details is not listed here.
5. Qubit based on Majorana Fermions: a Start
This section is a re-statement of our previously discussed Majorana case and demon-
strate why it’s probable to make quantum computation.
– 17 –
Ze-YangLi’sAssignment(Draft)
The idea to store quantum informa-
Figure 16: Top view of a 2D topological insulator,
contacted at the edge by two superconducting elec-
trodes separated by a magnetic tunnel junction. A
pair of Majorana fermions is bound by the super-
conducting and magnetic gaps. The tunnel split-
ting of the bound states depends on the supercon-
ducting phase difference φ, as indicated in the plot
[≈ cos(f/2)]. The crossing of the levels at φ = π is
protected by quasiparticle parity conservation. Fig-
ure from [10].
tion in Majorana fermions originates from
Alexei Kitaev[11], as in Figure 16. The
massless Dirac fermions now propagate
along a 1D edge state, again with the
spin pointing in the direction of motion.
(This is the helical edge state responsible
for the quantum spin Hall effect.) A Ma-
jorana fermion appears as a zero mode
at the interface between a superconduc-
tor (S) and a magnetic insulator (I). The
Figure shows two zero modes coupled by
tunneling in an SIS junction, forming a
two-level system (a qubit). The two states
|1〉and |0〉 of the qubit are distinguished
by the presence or absence of an unpaired
quasiparticle. For well-separated Majo-
ranas, with an exponentially small tun-
nel splitting, this is a nonlocal encoding
of quantum information: Each zero mode
by itself contains no information on the
quasiparticle parity. Dephasing of the
qubit is avoided by hiding the phase in
much the same way that one would hide
the phase of a complex number by separately storing the real and imaginary parts. The
complex Dirac fermion operator a = 1/2(γ1 + iγ2) of the qubit is split into two real Majo-
rana fermion operators. Whereas two Majoranas encode one qubit, 2n Majoranas encode
the quantum information of n qubits in 2n nearly degenerate states. Without these degen-
eracies, the adiabatic evolution of a state Ψ along a closed loop in parameter space would
simply amount to multiplication by a phase factor, Ψ→ eiαΨ, but now the operation may
result in multiplication by a unitary matrix, Ψ→ UΨ. Because matrix multiplications do
not commute, the order of the operations matters. This produces the non-Abelian statistics
discovered by Gregory Moore and Nicholas Read, in the context of the fractional quantum
Hall effect, and by Read and Dmitry Green, in the context of p-wave superconductors. The
adiabatic interchange (braiding) of two Majorana bound states is a non-Abelian unitary
transformation of the form
Ψ→ exp(iπ
4σz
)Ψ
Two interchanges return the Majoranas to their starting position, but the final state iσzΨ
is in general not equivalent to the initial stateΨ. Such an operation is called topological
because it is fully determined by the topology of the braiding; in particular, the coefficient
in the exponent is precisely π/4. This could be useful for a quantum computer, even though
not all unitary operations can be performed by the braiding of Majoranas. [10].
– 18 –
Ze-YangLi’sAssignment(Draft)
6. Perspective
Topological superconductor is still a very hot field, both experimentally and theoretically.
Hasan in Princeton recently released an experimental research about this particular subject
[12] which firstly go beyond ZBCP and DCP features to observe TSC2, and Ashvin pub-
lished one theoretical consideration paper [13] with fancy idea and fabulous mathematical
technique aimed at the well-known Non-abelian property of TSC.
TSC, on the one hand provides a fundamentally new way to store and manipulate
quantum information with possible applications in a quantum computer, inspires funda-
mental studies on non-local entanglement and the intrinsic property of space-time on the
other hand.
7. Acknowledgements
I would like to thank Prof. Jian Wang and Teaching Assistant Xi Zhang for giving me an
opportunity to receive such a fabulous lectures even as a undergraduate student. The slides
at course inspire me to think a lot and know much more than expected. I also benefited
when preparing the presentation for the course with Jia-Chen Yu and Shang-Jie Xue. I
also appreciate numerous help out of the course, especially Mr. Hong-Lie Ning who
gave me a great amount of help, both theoretically instruction and paper suggestion.
2PbTaSe2 material via ARPES method, but not so convincing yet.
– 19 –
Ze-YangLi’sAssignment(Draft)
Appendix
Appendix A 2-state Berry Phase Calculation
Consider a case where
H(R) = ε(R) + R · σ (Ap-1)
Straightforwardly, the energy levels are E± = ε(R)±√
d · d. Employ spherical coor-
dinates and then d = d(sin θ cosφ, sin θ sinφ, cos θ). Hence the two state is
|+〉 =
(cos(θ2
)e−iφ
sin(θ2
) ), |−〉 =
(sin(θ2
)e−iφ
− cos(θ2
) ) (Ap-2)
which are orthogonal and normalized. Two Berry vector potential and corresponding Berry
curvature are
Aθ = 0, Aφ = sin2
(θ
2
), Fθφ = ∂θAφ − ∂φAθ =
sin θ
2(Ap-3)
Immediately,
V−i = εijkFij, ⇒ V− =d
d3
like a monopole in R parameter space of strength. An integral over a loop that its surface
across this monopole gives nontrivial Chern number contribution. It’s the most important
concept and the simples example for Chern number calculating for
γ(C) =
∫∫cdS ·V
This can also be applied to consider a spin in a magnetic field where Hamiltonian is
given by H(B) = B · S. The Berry vector curvature is
Vn(B) = nB
B3, n = −S, · · · , S
and that explains the common arguing that a electron (or fermion) would turn back to
itself not with a 2π rotation but a 4π rotation.
Appendix I
Ze-YangLi’sAssignment(Draft)
References
[1] B Andrei Bernevig. Topological insulators and topological superconductors. Princeton
University Press, 2013.
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